The derivation of explicit exprEssions for the effective dielectric tensor to be utilized in the dispersion relation for weakly inhomogeneous Plasmas is discussed. The general exprEssions obtained are useful for situations with simultaneous existence of weak inhomogeneities in density and magnetic field. The particular case of a Maxwellian distribution in velocity space for the electron population is discussed, and relatively compact exprEssions for the dielectric tensor are obtained which depend on the inhomogeneous Plasma dispersion function introduced by [Gaelzer et al., Phys. Rev. E55, 5859 (1997)] and ultimately on the well-known Fried-Conte function and its derivatives.